Theorem efglem 17952

Description: Lemma for efgval 17953. (Contributed by Mario Carneiro, 27-Sep-2015.)

Hypothesis

Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))

Assertion

Ref	Expression
efglem	⊢ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))

Distinct variable groups:   𝑦,𝑟,𝑧,𝑛,𝑥,𝑊   𝑛,𝐼,𝑟,𝑥,𝑦,𝑧

Proof of Theorem efglem

Step	Hyp	Ref	Expression
1		xpider 7705	. 2 ⊢ (𝑊 × 𝑊) Er 𝑊
2		simpll 786	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝑥 ∈ 𝑊)
3		efgval.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
4		fviss 6166	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
5	3, 4	eqsstri 3598	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2𝑜)
6	5, 2	sseldi 3566	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
7		opelxpi 5072	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2𝑜))
8	7	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2𝑜))
9		2oconcl 7470	. . . . . . . . . 10 ⊢ (𝑧 ∈ 2𝑜 → (1𝑜 ∖ 𝑧) ∈ 2𝑜)
10		opelxpi 5072	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐼 ∧ (1𝑜 ∖ 𝑧) ∈ 2𝑜) → ⟨𝑦, (1𝑜 ∖ 𝑧)⟩ ∈ (𝐼 × 2𝑜))
11	9, 10	sylan2 490	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜) → ⟨𝑦, (1𝑜 ∖ 𝑧)⟩ ∈ (𝐼 × 2𝑜))
12	11	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → ⟨𝑦, (1𝑜 ∖ 𝑧)⟩ ∈ (𝐼 × 2𝑜))
13	8, 12	s2cld 13466	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩ ∈ Word (𝐼 × 2𝑜))
14		splcl 13354	. . . . . . 7 ⊢ ((𝑥 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2𝑜))
15	6, 13, 14	syl2anc 691	. . . . . 6 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2𝑜))
16	3	efgrcl 17951	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
17	16	simprd 478	. . . . . . 7 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2𝑜))
18	17	ad2antrr 758	. . . . . 6 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝑊 = Word (𝐼 × 2𝑜))
19	15, 18	eleqtrrd 2691	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ∈ 𝑊)
20		brxp 5071	. . . . 5 ⊢ (𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ↔ (𝑥 ∈ 𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ∈ 𝑊))
21	2, 19, 20	sylanbrc 695	. . . 4 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))
22	21	ralrimivva 2954	. . 3 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) → ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))
23	22	rgen2 2958	. 2 ⊢ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)
24		fvex 6113	. . . . 5 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ∈ V
25	3, 24	eqeltri 2684	. . . 4 ⊢ 𝑊 ∈ V
26	25, 25	xpex 6860	. . 3 ⊢ (𝑊 × 𝑊) ∈ V
27		ereq1 7636	. . . 4 ⊢ (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
28		breq 4585	. . . . . 6 ⊢ (𝑟 = (𝑊 × 𝑊) → (𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ↔ 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
29	28	2ralbidv 2972	. . . . 5 ⊢ (𝑟 = (𝑊 × 𝑊) → (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ↔ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
30	29	2ralbidv 2972	. . . 4 ⊢ (𝑟 = (𝑊 × 𝑊) → (∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ↔ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
31	27, 30	anbi12d 743	. . 3 ⊢ (𝑟 = (𝑊 × 𝑊) → ((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)) ↔ ((𝑊 × 𝑊) Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))))
32	26, 31	spcev 3273	. 2 ⊢ (((𝑊 × 𝑊) Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)) → ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)))
33	1, 23, 32	mp2an 704	1 ⊢ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩))

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ⟨cop 4131  ⟨cotp 4133   class class class wbr 4583   I cid 4948   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   Er wer 7626  0cc0 9815  ...cfz 12197  #chash 12979  Word cword 13146   splice csplice 13151  ⟨“cs2 13437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-s2 13444

This theorem is referenced by:  efgval  17953  efger  17954